The Antimagicness of Super (a, d) - P2⊵̇H Total Covering on Total Comb Graphs

Abstract

A graph can be constructed in several ways. One of them is by operating two or more graphs. The resulting graphs will be a new graph which has certain characteristics. One of the latest graph operations is total comb of two graphs. Let L, H be a finite collection of nontrivial, simple and undirected graphs. The total comb product is a graph obtained by taking one copy of L and |V(L)| + |E(L)| copies of H and grafting the i-th copy of H at the vertex o and edge uv to the i-th vertex and edge of L. The graph G

˙ H-antimagic total graph if there exists a bijective function f : V(G) ∪ E(G) →{1, 2,...,|V(G)| + |E(G)|} such that for all subgraphs isomorphic to P is said to be an (a, d)-P 2 2

˙ H, the total P 2

˙ H-weights W(P 2

˙ H) =

v∈V(P 2

˙ H) f (v) +

f (e) form an arithmetic sequence. An (a , d)-P

˙ H-antimagic total covering f is called super when the smallest labels appear in the vertices. By using partition technique has been proven that the graph G = L ˙ H admits a super (a, d)-P 2

˙ H antimagic total labeling with different value d = d ∗ + d ∗ (d v 1 + d e 1 ) + d v 2 + d e 2 + 1.